You remember those half-angle formulae

sin 2A = 2 sin A cos A

cos 2A = (cos A)^2 - (sin A)^2

If you are happy using sine of small x is x and cosine of small x is 1 as your base cases, you can write sine and cosine as mutually recursive functions:

         (defun sin (z)
           (if (small z)
               z
               (* 2 
                  (sin (half z))
                  (cos (half z)))))

         (defun cos (z)
           (if (small z)
               1
               (- (square (cos (half z)))
                  (square (sin (half z))))))

(sin pi) => 6.167817939221069d-7 quite close to zero (cos pi) => -1.0012055113842453d0 you can get this much closer to -1 by using 1-x^2/2 as your base case.

It had never occurred to me to do exponential the same way, as

         (defun exp (z)
           (if (small z)
               (+ 1 z)
               (square (exp (half z)))))

So it is a bit of a shock to try it and see it work just fine for complex numbers

(exp (complex 0 pi)) => #C(-1.0012055113842453d0 6.167817939221069d-7)